Let
C
(
−
d
)
C( - d)
denote the Gauss Class Group of quadratic forms of a negative discriminant
−
d
- d
(or equivalently, the class group of the imaginary quadratic field
Q
(
−
d
)
Q(\sqrt { - d} )
). We give a rigorous proof that there exists a Las Vegas algorithm that will compute the structure of
C
(
−
d
)
C( - d)
with an expected running time of
L
(
d
)
2
+
o
(
1
)
L{(d)^{\sqrt 2 + o(1)}}
bit operations, where
L
(
d
)
=
exp
(
log
d
log
log
d
)
L(d) = {\text {exp}}(\sqrt {\log \,d\;\log \,\log \,d} )
. Thus, of course, also includes the computation of the class number
h
(
−
d
)
h( - d)
, the cardinality of
C
(
−
d
)
C( - d)
.